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Theorem 0ss 3371
Description: The null set is a subset of any class. Part of Exercise 1 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
0ss  |-  (/)  C_  A

Proof of Theorem 0ss
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 noel 3337 . . 3  |-  -.  x  e.  (/)
21pm2.21i 620 . 2  |-  ( x  e.  (/)  ->  x  e.  A )
32ssriv 3071 1  |-  (/)  C_  A
Colors of variables: wff set class
Syntax hints:    e. wcel 1465    C_ wss 3041   (/)c0 3333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-dif 3043  df-in 3047  df-ss 3054  df-nul 3334
This theorem is referenced by:  ss0b  3372  ssdifeq0  3415  sssnr  3650  ssprr  3653  uni0  3733  int0el  3771  0disj  3896  disjx0  3898  tr0  4007  0elpw  4058  exmidsssn  4095  fr0  4243  elnn  4489  rel0  4634  0ima  4869  fun0  5151  f0  5283  oaword1  6335  0domg  6699  nnnninf  6991  exmidfodomrlemim  7025  sum0  11112  ennnfonelemj0  11825  ennnfonelemkh  11836  0opn  12084  baspartn  12128  0cld  12192  ntr0  12214  bdeq0  12961  bj-omtrans  13050  el2oss1o  13084  nninfsellemsuc  13104
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